Multiobjective Optimization Athens 2005

1. Multiobjective OptimizationAthens 2005 Department of Architecture and Technology Universidad Politécnica de Madrid Santiago González Tortosa

2. Contents • Introduction • Multiobjective Optimization • MO Non-Heuristic • Linear • Nonlinear • Handling Constraints Techniques • EMOO • Using Constraints in EMOO • Conclusions • References

3. Introduction • Optimization Problem • Find a solution in the feasible region which has the minimum (or maximum) value of the objective function • Possibilities • Unique objective (function) • Multiobjective • multiple optimal solutions • selection by preference function • Solving: • Non-Heuristic (deterministic) • Heuristic

4. Multiobjective Optimization • Find a solution that: • Minimize (objectives) f(x) = (f1(x), f2(x), ..., fn(x)) • Subject to (constraints) g(x) = (g1(x), g2(x), ... ,gm(x)) ≤ 0 • x = (x1, …, xn) • f, g linear/nonlinear functions

5. Multiobjective Optimization • Δ (The searching space): set of all possible solutions of x. • Ð (The feasible space): set of all solutions that satisfy all the constraints.

6. Multiobjective Optimization • Pareto Optimal • x℮Ðis said to be Pareto Optimal if there does not exist another solution x’ ℮Ð that • fi(x) = fi(x’) i = 1, …, m • fi(x) < fi(x’) i = 1, …, m at least one i. • x solution dominate x’ solution • Pareto Front • The maximal set of non-dominated feasible solutions.

7. MO Non-Heuristic • Multiobjective Optimization • Linear (f and g) • Multiobjective Simplex Method • Techniques: • Multiparametric Decomposition (weights a priori) • Fractional Program (ratio objectives) • Goals Program (goal deviations) • Nonlinear (f or g) • Compromise Programming • Ideal Solution for each objective function • Distance between solutions • Objective weights • Compromised Solution • Compensation of objectives • Competitive Objectives

8. MO Non-Heuristic

9. Constraints-Handling Techniques • Penalty Function • Very easy  • Depends on problem  • Problems with strong constraints  • Repair Heuristic • Useful when it’s difficult to find feasible problems  • Depends on problem  • Separation between objectives & constraints • No depends on problem  • Extend to multiobjective optimization problems  • Hybrid Methods • Use of numerical optimization problem  • Excessive computational cost  • Others

10. EMOO • Evolutionary MultiObjective Optimization • Techniques: • A priori • Preferences before executing • Reduce the problem to a unique objective • Unique solution • A posteriori • Preferences after executing • Multiple solutions • Methods: • Non-based on Pareto Optimal concept • based on Pareto Optimal concept • Non-elitist • elitist

11. EMOO • A posteriori: Non-based on Pareto Optimal concept • VEGA algorithm (Vector Evaluated Genetic Algorithm) • k objectives, population size N • Subpopulations size N/k • Calculate fitness function and select t best individuals (create new subpopulation) • Shuffle all subpopulations • Apply GA operators and create new populations of size N • Speciation Problem: select individuals depending on 1 objective only

12. EMOO • A posteriori: non-elitist based on Pareto Optimal concept • MOGA algorithm(MultiObjective Genetic Algorithm) • range (x) = 1 + p(x) (p(x) number of individuals that dominate it) • Sorting by minimal range • Create a dummy fitness (lineal or non-linear) and calculate (interpolate) depending on individual range • Select t best individuals (niches) • Apply GA operators and create new population • Others: NSGA, NPGA, …

13. EMOO • A posteriori: elitist based on Pareto Optimal concept • NSGA-II algorithm (Non-dominated Sorting Genetic Algorithm) • Population P (size N) • Create new population P’ (size N) using GA operators • Merge both populations and create new Population R (size 2N) • Sort by range of domination • Select t individuals (tournament & niches) and create a new population R’ • Others: DPGA, PESA, PAES, MOMGA, …

14. Using Constraints in EMOO • Eliminate non-feasible solutions • Use Penalty functions • Separate solutions feasible and non-feasible • Define problem with Goals

15. Conclusions • Multiobjective Optimization • Non-Heuristic • Multiobjective Simplex Method (Linear) or Compromised Programming (Non-Linear) • Using Constraints • Goals, Penalties, Weights… • Heuristic • Using GA (EMOO) • A priori (unique objective) • A posteriori • Using GA and Constraints

16. References • Gracia Sánchez Carpena. Diseño y Evaluación de Algoritmos Evolutivos Multiobjetivo en Optimización y Modelación Difusa, PhD Thesis, Departamento de Ingeniería de la Información y las Comunicaciones, Universidad de Murcia, Murcia, Spain, November, 2002 (in Spanish). • Carlos A. Coello Coello, David A. Van Veldhuizen and Gary B. Lamont, Evolutionary Algorithms for Solving Multi-Objective Problems, Kluwer Academic Publishers, New York, March 2002, ISBN 0-3064-6762-3. David A. Van Veldhuizen. Multiobjective Evolutionary algorithms: Classifications, Analyses, and New Innovations. PhD thesis, Department of Electrical and Computer Engineering. Graduate School of Engineering. Air Force Institute of Technology, Wright-Patterson AFB, Ohio, May 1999. • J. David Schaffer. Multiple Objective Optimization with Vector Evaluated Genetic Algorithms. PhD thesis, Vanderbilt University, 1984. • Tadahiko Murata. Genetic Algorithms for Multi-Objective Optimization. PhD thesis, Osaka Prefecture University, Japan, 1997. • Kalyanmoy Deb, Associate Member, IEEE, Amrit Pratap, Sameer Agarwal, and T. Meyarivan A Fast and Elitist Multiobjective Genetic Algorithm:NSGA-II • Kaisa Miettinen Nonlinear Multiobjective Optimization Kluwer Academic Publishers, Boston, 1999

17. Multiobjective OptimizationAthens 2005 Department of Architecture and Technology Universidad Politécnica de Madrid Santiago González Tortosa